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In fact the relation that we uncovered in [16] between the Hopf algebra of Feynman graphs and the Hopf algebra of coordinates on the group of formal diffeomorphisms of the dimensionless coupling constants of the theory allows to prove the following result which for simplicity deals with the case of a single dimensionless coupling constant. Let the unrenormalized effective coupling constant g eff (ε) be viewed as a formal power series in g and let g eff (ε) = g eff+ (ε) (g eff− (ε)) −1 be its (opposite) Birkhoff decomposition in the group of formal diffeomorphisms. Then the loop g eff− (ε) is the bare coupling constant and g eff+ (0) is the renormalized effective coupling. This allows, using the relation between the Birkhoff decomposition and the classification of holomorphic bundles, to encode geometrically the operation of renormalization. It also signals a very clear analogy between the renormalization group as an ”ambiguity” group of physical theories and the missing Galois theory at Archimedian places alluded to above. We refer to [14] for a detailed account of this analogy. I References [1] M.F. Atiyah, Global theory of elliptic operators, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969) University of Tokyo press, Tokyo, (1970), 21-30. [2] P. Baum and A. Connes, Geometric K-theory for Lie groups and foliations. Preprint IHES (M/82/), 1982, to appear in l’Enseignement Mathematique, t.46 (2000), 1-35. [3] L.G. Brown, R.G. Douglas and P.A. Fillmore, Extensions of C ∗ -algebras and K-homology, Ann. of Math. 2, 105 (1977), 265-324. [4] A. Chamseddine and A. Connes: Universal formulas for noncommutative geometry actions, Phys. Rev. Letters 77 24 (1996), 4868-4871. [5] J. Chabert, S. Echterhoff, R. Nest, The Connes-Kasparov conjecture for almost connected groups, MathQA/0110130. [6] A. Connes, Une classification des facteurs de type III, Ann. Sci. Ecole Norm. Sup., 6, n. 4 (1973), 133-252. [7] A. Connes, Classification of injective factors, Ann. of Math., 104, n. 2 (1976), 73-115. [8] Connes, A., C ∗ algèbres et géométrie differentielle. C.R. Acad. Sci. Paris, Ser. A-B 290 (1980). [9] A. Connes, Noncommutative differential geometry. Part I: The Chern character in K-homology, Preprint IHES (M/82/53), 1982; Part II: de Rham homology and noncommutative algebra, Preprint IHES (M/83/19), 1983. Noncommutative differential geometry, Inst. Hautes Etudes Sci. Publ. Math. 62 (1985), 257-360. [10] Connes, A., : Cohomologie cyclique et foncteur Ext n . C.R. Acad. Sci. Paris, Ser.I Math 296 (1983). 22
[11] A. Connes: Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras, (Kyoto, 1983), pp. 52-144, Pitman Res. Notes in Math. 123 Longman, Harlow (1986). [12] A. Connes: Noncommutative geometry, Academic Press (1994). [13] A. Connes, Noncommutative Geometry and the Riemann Zeta Function, invited lecture in IMU 2000 volume, to appear. [14] A. Connes, Symétries Galoisiennes et renormalisation. Séminaire H. Poincaré (2002). [15] A. Connes, M. Douglas, A. Schwarz, Noncommutative geometry and Matrix theory: compactification on tori, J. High Energy Physics, 2 (1998). [16] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem. hep-th/9912092. Renormalization in quantum field theory and the Riemann-Hilbert problem II: The β function, diffeomorphisms and the renormalization group. hep-th/0003188. [17] A. Connes and H. Moscovici: Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), 345-388. [18] A. Connes and H. Moscovici: The local index formula in noncommutative geometry, GAFA, 5 (1995), 174-243. [19] Connes, A. and Moscovici, H., Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem, Commun. Math. Phys. 198, (1998) 199-246. [20] A. Connes and M. Rieffel, Yang-Mills for noncommutative two tori, Operator algebras and mathematical physics, (Iowa City, Iowa, 1985), pp. 237-266; Contemp. Math. Oper. Algebra Math. Phys. 62, Amer. Math. Soc., Providence, RI, 1987. [21] A. Connes and M. Takesaki, The flow of weights on factors of type III, Tohoku Math. J., 29 (1977), 473-575. [22] U. Haagerup, Connes’ bicentralizer problem and uniqueness of the injective factor of type III 1 , Acta Math., 158 (1987), 95-148. [23] D. Kastler, The Dirac operator and gravitation, Commun. Math. Phys. 166 (1995), 633-643. [24] D. Kastler, Noncommutative geometry and fundamental physical interactions: The lagrangian level, Journal. Math. Phys. 41 (2000), 3867-3891. [25] G.G. Kasparov, The operator K-functor and extensions of C ∗ algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 571-636; Math. USSR Izv. 16 (1981), 513-572. [26] W. Krieger, On ergodic flows and the isomorphism of factors, Math. Ann., 223 (1976), 19-70. 23
Page 1 and 2: NONCOMMUTATIVE GEOMETRY Alain Conne
Page 3 and 4: topological spaces. They are given
Page 5 and 6: i.e. X is the quotient of S 1 by th
Page 7 and 8: and the δ j are still derivations
Page 9 and 10: What I mean is that it admits a god
Page 11 and 12: Lie groups. He also gave the first
Page 13 and 14: Of course such a dictionary that tr
Page 15 and 16: In practice the choice of the limit
Page 17 and 18: and is given by where D is the Dira
Page 19 and 20: There is a very important test of o
Page 21: in [18], encoded by a spectral trip

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